5.02 Function notation

We have seen that functions are special types of relations. When dealing with functions we often use function notation to emphasise that we are dealing with a special case, to highlight the independent and dependent variable and for ease of notation with substitution and calculus.

When we are writing in function notation, instead of writing "$$y=", we write "$$f(x)=". This can be interpreted as "$$f is a function of the variable $$x" and read as "$$f of $$x". Common letters to use for general functions are lower case $$f, $$g and $$h. However, any letter can be used and we can use variables that have meaning in context. 

Exploration

Instead of $$y=2x+1 we could write $$f(x)=2x+1, here $$f is a function of the variable $$x which follows the rule double $$x and add $$1.

$$P(t)=200×0.8t, here population is a function of time.

$$H(d)=30−2d−30d2, here height is a function of distance.

These are all examples of just a different way to write '$$y= …' notation as a function. The letter in the bracket on the left is the input and the right hand side gives us a rule for the output.

Function notation also allows for shorthand for substitution. If $$y=3x+2 and we write this in function notation as $$f(x)=3x+2, then the question 'what is the value of $$y when $$x is $$5?' can be asked simply as 'what is $$f(5)?' or 'Evaluate $$f(5).'

 

Worked example

If $$A(x)=x2+1 and $$Q(x)=x2+9x, find:

(a) $$A(5)

Think: This means we need to substitute $$5 in for $$x in the $$A(x) equation.

Do: 

$$A(5)	$$=	$$52+1
	$$=	$$26

 

(b) $$Q(6)

Think: This means we need to substitute $$6 in for $$x in the $$Q(x) equation.

Do:

$$Q(6)	$$=	$$62+9×6
	$$=	$$36+54
	$$=	$$90

 

(c) $$A(p)

Think: This means we need to substitute $$p in for $$x in the $$A(x) equation.

Do: 

$$A(p)

$$=

$$p2+1

 

Practice questions

Question 1

Question 2

Question 3

Question 4